Projectile Motion

views updated

PROJECTILE MOTION

CONCEPT

A projectile is any object that has been thrown, shot, or launched, and ballistics is the study of projectile motion. Examples of projectiles range from a golf ball in flight, to a curve ball thrown by a baseball pitcher to a rocket fired into space. The flight paths of all projectiles are affected by two factors: gravity and, on Earth at least, air resistance.

HOW IT WORKS

The effects of air resistance on the behavior of projectiles can be quite complex. Because effects due to gravity are much simpler and easier to analyze, and since gravity applies in more situations, we will discuss its role in projectile motion first. In most instances on Earth, of course, a projectile will be subject to both forces, but there may be specific cases in which an artificial vacuum has been created, which means it will only be subjected to the force of gravity. Furthermore, in outer space, gravity—whether from Earth or another body—is likely to be a factor, whereas air resistance (unless or until astronomers find another planet with air) will not be.

The acceleration due to gravity is 32 ft (9.8 m)/sec², usually expressed as "per second squared." This means that as every second passes, the speed of a falling object is increasing by 32 ft/sec (9.8 m). Where there is no air resistance, a ball will drop at a velocity of 32 feet per second after one second, 64 ft (19.5 m) per second after two seconds, 96 ft (29.4 m) per second after three seconds, and so on. When an object experiences the ordinary acceleration due to gravity, this figure is rendered in shorthand as g. Actually, the figure of 32 ft (9.8 m) per second squared applies at sea level, but since the value of g changes little with altitude—it only decreases by 5% at a height of 10 mi (16 km)—it is safe to use this number.

When a plane goes into a high-speed turn, it experiences much higher apparent g. This can be as high as 9 g, which is almost more than the human body can endure. Incidentally, people call these " g -forces," but in fact g is not a measure of force but of a single component, acceleration. On the other hand, since force is the product of mass multiplied by acceleration, and since an aircraft subject to a high g factor clearly experiences a heavy increase in net force, in that sense, the expression " g -force" is not altogether inaccurate.

In a vacuum, where air resistance plays no part, the effects of g are clearly demonstrated. Hence a cannonball and a feather, dropped into a vacuum at the same moment, would fall at exactly the same rate and hit bottom at the same time.

The Cannonball or the Feather? Air Resistance vs. Mass

Naturally, air resistance changes the terms of the above equation. As everyone knows, under ordinary conditions, a cannonball falls much faster than a feather, not simply because the feather is lighter than the cannonball, but because the air resists it much better. The speed of descent is a function of air resistance rather than mass, which can be proved with the following experiment. Using two identical pieces of paper—meaning that their mass is exactly the same—wad one up while keeping the other flat. Then drop them. Which one lands first? The wadded piece will fall faster and land first, precisely because it is less air-resistant than the sail-like flat piece.

Now to analyze the motion of a projectile in a situation without air resistance. Projectile motion follows the flight path of a parabola, a curve generated by a point moving such that its distance from a fixed point on one axis is equal to its distance from a fixed line on the other axis. In other words, there is a proportional relationship between x and y throughout the trajectory or path of a projectile in motion. Most often this parabola can be visualized as a simple up-and-down curve like the shape of a domed roof. (The Gateway Arch in St. Louis, Missouri, is a steep parabola.)

Instead of referring to the more abstract values of x and y, we will separate projectile motion into horizontal and vertical components. Gravity plays a role only in vertical motion, whereas obviously, horizontal motion is not subject to gravitational force. This means that in the absence of air resistance, the horizontal velocity of a projectile does not change during flight; by contrast, the force of gravity will ultimately reduce its vertical velocity to zero, and this will in turn bring a corresponding drop in its horizontal velocity.

In the case of a cannonball fired at a 45° angle—the angle of maximum efficiency for height and range together—gravity will eventually force the projectile downward, and once it hits the ground, it can no longer continue on its horizontal trajectory. Not, at least, at the same velocity: if you were to thrust a bowling ball forward, throwing it with both hands from the solar plexus, its horizontal velocity would be reduced greatly once gravity forced it to the floor. Nonetheless, the force on the ball would probably be enough (assuming the friction on the floor was not enormous) to keep the ball moving in a horizontal direction for at least a few more feet.

There are several interesting things about the relationship between gravity and horizontal velocity. Assuming, once again, that air resistance is not a factor, the vertical acceleration of a projectile is g. This means that when a cannonball is at the highest point of its trajectory, you could simply drop another cannonball from exactly the same height, and they would land at the same moment. This seems counterintuitive, or opposite to common sense: after all, the cannonball that was fired from the cannon has to cover a great deal of horizontal space, whereas the dropped ball does not. Nonetheless, the rate of acceleration due to gravity will be identical for the two balls, and the fact that the ball fired from a cannon also covers a horizontal distance during that same period is purely incidental.

Gravity, combined with the first law of motion, also makes it possible (in theory at least) for a projectile to keep moving indefinitely. This actually does take place at high altitudes, when a satellite is launched into orbit: Earth's gravitational pull, combined with the absence of air resistance or other friction, ensures that the satellite will remain in constant circular motion around the planet. The same is theoretically possible with a cannonball at very low altitudes: if one could fire a ball at 17,700 MPH (28,500 k/mh), the horizontal velocity would be great enough to put the ball into low orbit around Earth's surface.

The addition of air resistance or airflow to the analysis of projectile motion creates a number of complications, including drag, or the force that opposes the forward motion of an object in airflow. Typically, air resistance can create a drag force proportional to the squared value of a projectile's velocity, and this will cause it to fall far short of its theoretical range.

Shape, as noted in the earlier illustration concerning two pieces of paper, also affects air resistance, as does spin. Due to a principle known as the conservation of angular momentum, an object that is spinning tends to keep spinning; moreover, the orientation of the spin axis (the imaginary "pole" around which the object is spinning) tends to remain constant. Thus spin ensures a more stable flight.

REAL-LIFE APPLICATIONS

Bullets on a Straight Spinning Flight

One of the first things people think of when they hear the word "ballistics" is the study of gunfire patterns for the purposes of crime-solving. Indeed, this application of ballistics is a significant part of police science, because it allows law-enforcement investigators to determine when, where, and how a firearm was used. In a larger sense, however, the term as applied to firearms refers to efforts toward creating a more effective, predictable, and longer bullet trajectory.

From the advent of firearms in the West during the fourteenth century until about 1500, muskets were hopelessly unreliable. This was because the lead balls they fired had not been fitted to the barrel of the musket. When fired, they bounced erratically off the sides of the barrel, and this made their trajectories unpredictable. Compounding this was the unevenness of the lead balls themselves, and this irregularity of shape could lead to even greater irregularities in trajectory.

Around 1500, however, the first true rifles appeared, and these greatly enhanced the accuracy of firearms. The term rifle comes from the "rifling" of the musket barrels: that is, the barrels themselves were engraved with grooves, a process known as rifling. Furthermore, ammunition-makers worked to improve the production process where the musket balls were concerned, producing lead rounds that were more uniform in shape and size.

Despite these improvements, soldiers over the next three centuries still faced many challenges when firing lead balls from rifled barrels. The lead balls themselves, because they were made of a soft material, tended to become misshapen during the loading process. Furthermore, the gunpowder that propelled the lead balls had a tendency to clog the rifle barrel. Most important of all was the fact that these rifles took time to load—and in a situation of battle, this could cost a man his life.

The first significant change came in the 1840s, when in place of lead balls, armies began using bullets. The difference in shape greatly improved the response of rounds to aerodynamic factors. In 1847, Claude-Etienne Minié, a captain in the French army, developed a bullet made of lead, but with a base that was slightly hollow. Thus when fired, the lead in the round tended to expand, filling the barrel's diameter and gripping the rifling.

As a result, the round came out of the barrel end spinning, and continued to spin throughout its flight. Not only were soldiers able to fire their rifles with much greater accuracy, but thanks to the development of chambers and magazines, they could reload more quickly.

Curve Balls, Dimpled Golf Balls, and Other Tricks with Spin

In the case of a bullet, spin increases accuracy, ensuring that the trajectory will follow an expected path. But sometimes spin can be used in more complex ways, as with a curveball thrown by a baseball pitcher.

The invention of the curveball is credited to Arthur "Candy" Cummings, who as a pitcher for the Brooklyn Excelsiors at the age of 18 in 1867—an era when baseball was still very young—introduced a new throw he had spent several years perfecting. Snapping as he released the ball, he and the spectators (not to mention the startled batter for the opposing team) watched as the pitch arced, then sailed right past the batter for a strike.

The curveball bedeviled baseball players and fans alike for many years thereafter, and many dismissed it as a type of optical illusion. The debate became so heated that in 1941, both Life and Look magazines ran features using stop-action photography to show that a curveball truly did curve. Even in 1982, a team of researchers from General Motors (GM) and the Massachusetts Institute of Technology (MIT), working at the behest of Science magazine, investigated the curveball to determine if it was more than a mere trick.

In fact, the curveball is a trick, but there is nothing fake about it. As the pitcher releases the ball, he snaps his wrist. This puts a spin on the projectile, and air resistance does the rest. As the ball moves toward the plate, its spin moves against the air, which creates an airstream moving against the trajectory of the ball itself. The airstream splits into two lines, one curving over the ball and one curving under, as the ball sails toward home plate.

For the purposes of clarity, assume that you are viewing the throw from a position between third base and home. Thus, the ball is moving from left to right, and therefore the direction of airflow is from right to left. Meanwhile the ball, as it moves into the airflow, is spinning clockwise. This means that the air flowing over the top of the ball is moving in a direction opposite to the spin, whereas that flowing under it is moving in the same direction as the spin.

This creates an interesting situation, thanks to Bernoulli's principle. The latter, formulated by Swiss mathematician and physicist Daniel Bernoulli (1700-1782), holds that where velocity is high, pressure is low—and vice versa. Bernoulli's principle is of the utmost importance to aerodynamics, and likewise plays a significant role in the operation of a curveball. At the top of the ball, its clockwise spin is moving in a direction opposite to the airflow. This produces drag, slowing the ball, increasing pressure, and thus forcing it downward. At the bottom end of the ball, however, the clockwise motion is flowing with the air, thus resulting in higher velocity and lower pressure. As per Bernoulli's principle, this tends to pull the ball downward.

In the 60-ft, 6-in (18.4-m) distance that separates the pitcher's mound from home plate on a regulation major-league baseball field, a curve-ball can move downward by a foot (0.3048 m) or more. The interesting thing here is that this downward force is almost entirely due to air resistance rather than gravity, though of course gravity eventually brings any pitch to the ground, assuming it has not already been hit, caught, or bounced off a fence.

A curveball represents a case in which spin is used to deceive the batter, but it is just as possible that a pitcher may create havoc at home plate by throwing a ball with little or no spin. This is called a knuckleball, and it is based on the fact that spin in general—though certainly not the deliberate spin of a curveball—tends to ensure a more regular trajectory. Because a knuckleball has no spin, it follows an apparently random path, and thus it can be every bit as tricky for the pitcher as for the batter.

Golf, by contrast, is a sport in which spin is expected: from the moment a golfer hits the ball, it spins backward—and this in turn helps to explain why golf balls are dimpled. Early golf balls, known as featheries, were merely smooth leather pouches containing goose feathers. The smooth surface seemed to produce relatively low drag, and golfers were impressed that a well-hit feathery could travel 150-175 yd (137-160 m).

Then in the late nineteenth century, a professor at St. Andrews University in Scotland realized that a scored or marked ball would travel farther than a smooth one. (The part about St. Andrews may simply be golfing legend, since the course there is regarded as the birthplace of golf in the fifteenth century.) Whatever the case, it is true that a scored ball has a longer trajectory, again as a result of the effect of air resistance on projectile motion.

Airflow produces two varieties of drag on a sphere such as a golf ball: drag due to friction, which is only a small aspect of the total drag, and the much more significant drag that results from the separation of airflow around the ball. As with the curveball discussed earlier, air flows above and below the ball, but the issue here is more complicated than for the curved pitch.

Airflow comes in two basic varieties: laminar, meaning streamlined; or turbulent, indicating an erratic, unpredictable flow. For a jet flying through the air, it is most desirable to create a laminar flow passing over its airfoil, or the curved front surface of the wing. In the case of the golf ball, however, turbulent flow is more desirable.

In laminar flow, the airflow separates quickly, part of it passing over the ball and part passing under. In turbulent flow, however, separation comes later, further back on the ball. Each form of air separation produces a separation region, an area of drag that the ball pulls behind it (so to speak) as it flies through space. But because the separation comes further back on the ball in turbulent flow, the separation region itself is narrower, thus producing less drag.

Clearly, scoring the ball produced turbulent flow, and for a few years in the early twentieth century, manufacturers experimented with designs that included squares, rectangles, and hexagons. In time, they settled on the dimpled design known today. Golf balls made in Britain have 330 dimples, and those in America 336; in either case, the typical drive distance is much, much further than for an unscored ball—180-250 yd (165-229 m).

Powered Projectiles: Rockets and Missiles

The most complex form of projectile widely known in modern life is the rocket or missile. Missiles are unmanned vehicles, most often used in warfare to direct some form of explosive toward an enemy. Rockets, on the other hand, can be manned or unmanned, and may be propulsion vehicles for missiles or for spacecraft. The term rocket can refer either to the engine or to the vehicle it propels.

The first rockets appeared in China during the late medieval period, and were used unsuccessfully by the Chinese against Mongol invaders in the early part of the thirteenth century. Europeans later adopted rocketry for battle, as for instance when French forces under Joan of Arc used crude rockets in an effort to break the siege on Orleans in 1429.

Within a century or so, however, rocketry as a form of military technology became obsolete, though projectile warfare itself remained as effective a method as ever. From the catapults of Roman times to the cannons that appeared in the early Renaissance to the heavy artillery of today, armies have been shooting projectiles against their enemies. The crucial difference between these projectiles and rockets or missiles is that the latter varieties are self-propelled.

Only around the end of World War II did rocketry and missile warfare begin to reappear in new, terrifying forms. Most notable among these was Hitler's V-2 "rocket" (actually a missile), deployed against Great Britain in 1944, but fortunately developed too late to make an impact. The 1950s saw the appearance of nuclear war-heads such as the ICBM (intercontinental ballistic missile). These were guided missiles, as opposed to the V-2, which was essentially a huge self-propelled bullet fired toward London.

More effective than the ballistic missile, however, was the cruise missile, which appeared in later decades and which included aerodynamic structures that assisted in guidance and maneuvering. In addition to guided or unguided, ballistic or aerodynamic, missiles can be classified in terms of source and target: surface-to-surface, air-to-air, and so on. By the 1970s, the United States had developed an extraordinarily sophisticated surface-to-air missile, the Stinger. Stingers proved a decisive factor in the Afghan-Soviet War (1979-89), when U.S.-supplied Afghan guerrillas used them against Soviet aircraft.

In the period from the late 1940s to the late 1980s, the United States, the Soviet Union, and other smaller nuclear powers stockpiled these warheads, which were most effective precisely because they were never used. Thus, U.S. President Ronald Reagan played an important role in ending the Cold War, because his weapons buildup forced the Soviets to spend money they did not have on building their own arsenal. During the aftermath of the Cold War, America and the newly democratized Russian Federation worked to reduce their nuclear stockpiles. Ironically, this was also the period when sophisticated missiles such as the Patriot began gaining widespread use in the Persian Gulf War and later conflicts.

Certain properties unite the many varieties of rocket that have existed across time and space—including the relatively harmless fireworks used in Fourth of July and New Year's Eve celebrations around the country. One of the key principles that makes rocket propulsion possible is the third law of motion. Sometimes colloquially put as "For every action, there is an equal and opposite reaction," a more scientifically accurate version of this law would be: "When one object exerts a force on another, the second object exerts on the first a force equal in magnitude but opposite in direction."

In the case of a rocket, propulsion comes by expelling fluid—which in scientific terms can mean a gas as well as a liquid—from its rear. Most often this fluid is a mass of hot gases produced by a chemical reaction inside the rocket's body, and this backward motion creates an equal and opposite reaction from the rocket, propelling it forward.

Before it undergoes a chemical reaction, rocket fuel may be either in solid or liquid form inside the rocket's fuel chamber, though it ends up as a gas when expelled. Both solid and liquid varieties have their advantages and disadvantages in terms of safety, convenience, and efficiency in lifting the craft. Scientists calculate efficiency by a number of standards, among them specific impulse, a measure of the mass that can be lifted by a particular type of fuel for each pound of fuel consumed (that is, the rocket and its contents) per second of operation time. Figures for specific impulse are rendered in seconds.

A spacecraft may be divided into segments or stages, which can be released as specific points along the flight in part to increase specific impulse. This was the case with the Saturn 5 rockets that carried astronauts to the Moon in the period 1969-72, but not with the varieties of space shuttle that have flown regular missions since 1981.

The space shuttle is essentially a hybrid of an airplane and rocket, with a physical structure more like that of an aircraft but with rocket power. In fact, the shuttle uses many rockets to maximize efficiency, utilizing no less than 67 rockets—49 of which run on liquid fuel and the rest on solid fuel—at different stages of its flight.

WHERE TO LEARN MORE

"Aerodynamics in Sports Equipment." K8AIT Principles of Aeronautics—Advanced (Web site). <http://muttley.ucdavis.edu/Book/Sports/advanced/index.html> (March 2, 2001).

Asimov, Isaac. Exploring Outer Space: Rockets, Probes, and Satellites. Revisions and updates by Francis Reddy. Milwaukee, WI: G. Stevens, 1995.

How in the World? Pleasantville, N.Y.: Reader's Digest, 1990.

"Interesting Properties of Projectile of Motion" (Web site). <http://www.phy.ntnu.edu.tw/~hwang/projectile3/projectile3.html> (March 2, 2001).

JBM Small Arms Ballistics (Web site). <http://roadrunner.com/~jbm/index_rgt.html> (March 2, 2001).

The Physics of Projectile Motion (Web site). <http://library.thinkquest.org/2779/> (March 2, 2001).

Richardson, Hazel. How to Build a Rocket. Illustrated by Scoular Anderson. New York: F. Watts, 2001.

KEY TERMS

ACCELERATION:

A change in velocity over a given time period.

AERODYNAMIC:

Relating to airflow.

BALLISTICS:

The study of projectil emotion.

DRAG:

The force that opposes the forward motion of an object in airflow. In most cases, its opposite is lift.

FIRST LAW OF MOTION:

A principle, formulated by Sir Isaac Newton (1642-1727), which states that an object at rest will remain at rest, and an object in motion will remain in motion, at a constant velocity unless or until outside forces act upon it.

FRICTION:

Any force that resists the motion of body in relation to another with which it is in contact.

INERTIA:

The tendency of an object in motion to remain in motion, and of an object at rest to remain at rest.

LAMINAR:

A term describing a streamlined flow, in which all particles move at the same speed and in the same direction. Its opposite is turbulent flow.

LIFT:

An aerodynamic force perpendicular to the direction of the wind. In most cases, its opposite is drag.

MASS:

A measure of inertia, indicating the resistance of an object to a change in its motion—including a change in velocity.

PARABOLA:

A curve generated by a point moving such that its distance from a fixed point on one axis is equal to its distance from a fixed line on the other axis. As a result, between any two points on the parabola there is a proportional relationship between x and y values.

PROJECTILE:

Any object that has been thrown, shot, or launched.

SPECIFIC IMPULSE:

A measure of rocket fuel efficiency—specifically, the mass that can be lifted by a particular type of fuel for each pound of fuel consumer (that is, the rocket and its contents) per second of operation time. Figures for specific impulse are rendered in seconds.

SPEED:

The rate at which the position of an object changes over a given period of time. Unlike velocity, direction is not a component of speed.

THIRD LAW OF MOTION:

A principle, which like the first law of motion was formulated by Sir Isaac Newton. The third law states that when one object exerts a force on another, the second object exerts on the first a force equal in magnitude but opposite in direction.

TRAJECTORY:

The path of a projectile in motion, a parabola upward and acrossspace.

TURBULENT:

A term describing a highly irregular form of flow, in which a fluid is subject to continual changes in speed and direction. Its opposite is laminar flow.